This is probably a layman's type of question, but since numbers are a symbolic representation of some sort of unit, is not the number play only bringing into focus the dysfunction of objectifying subjectivity?
There's no "number play" or actual problems with the paradox existing and there's no real problem in objectifying things in math. There are rules and axioms theories build on and using these we can build who logical worlds, ie "a+b=b+a" only using that axiom and logic you can come to countless other conclusions.
Are things ACTUALLY discrete or only so for our conceptual necessity?
By discrete I'm assuming you mean discrete in the mathematical sense (discrete vs continuous). If you mean in the physical world then the answer is depends we have countless models and theories for how physical reality is structured some of them are discrete and some are continuous but often times you can convert between the two on a conceptual and mathematical level.
Is the comprehension of infinity and/or a continuum made impossible by the finite capacity of the theorist?
Definitely not, although math takes a lot of creativity to be done at a high level.